Going back to the poles of the mass-spring system in Eq.
(F.36),
we see that, as the imaginary part of the two poles,
, approach zero, they come together at
to create a
repeated pole. The same thing happens at
since
both poles go to ``the point at infinity''.

It is a well known fact from linear systems theory that two poles at
the same point
in the
plane can correspond to an
impulse-response component of the form
, in addition
to the component
produced by a single pole at
. In the discrete-time case, a double pole at
can
give rise to an impulse-response component of the form
.
This is the fundamental source of the linearly growing internal states
of the wave digital sine oscillator at dc and
. It is
interesting to note, however, that such modes are always
unobservable at any physical output such as the massforce or spring force that is not actually linearly growing.